# Difference between revisions of "Nilpotent group"

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$$G=A_1\supseteq A_2\supseteq\ldots\supseteq A_{k+1}=\{1\}$$ | $$G=A_1\supseteq A_2\supseteq\ldots\supseteq A_{k+1}=\{1\}$$ | ||

− | such that every quotient $A_i/A_{i+1}$ lies in the centre of $G/A_{i+1}$ (a so-called central series). The length of a shortest central series of a nilpotent group is called its class (or degree of nilpotency). In any nilpotent group the lower (and upper) central series (see [[Subgroup series|Subgroup series]]) breaks off at the trivial subgroup (the group itself), and their lengths are equal to the nilpotency class of the group. | + | such that every quotient $A_i/A_{i+1}$ lies in the centre of $G/A_{i+1}$ (a so-called ''[[Central series of a group|central series]]''). The length of a shortest central series of a nilpotent group is called its class (or degree of nilpotency). In any nilpotent group the lower (and upper) central series (see [[Subgroup series|Subgroup series]]) breaks off at the trivial subgroup (the group itself), and their lengths are equal to the nilpotency class of the group. |

The finite nilpotent groups are exhausted by direct products of $p$-groups, that is, groups of orders $p^k$, where $p$ is a prime number. In any nilpotent group the elements of finite order form a subgroup, the quotient group by which is torsion free. The finitely-generated torsion-free nilpotent groups are exhausted by the groups of integral triangular matrices with 1's along the main diagonal, and their subgroups. Every finitely-generated torsion-free nilpotent group can be approximated by a finite $p$-group for every prime $p$. Finitely-generated nilpotent groups are polycyclic groups (cf. [[Polycyclic group|Polycyclic group]]) and, moreover, have a central series with cyclic factors. | The finite nilpotent groups are exhausted by direct products of $p$-groups, that is, groups of orders $p^k$, where $p$ is a prime number. In any nilpotent group the elements of finite order form a subgroup, the quotient group by which is torsion free. The finitely-generated torsion-free nilpotent groups are exhausted by the groups of integral triangular matrices with 1's along the main diagonal, and their subgroups. Every finitely-generated torsion-free nilpotent group can be approximated by a finite $p$-group for every prime $p$. Finitely-generated nilpotent groups are polycyclic groups (cf. [[Polycyclic group|Polycyclic group]]) and, moreover, have a central series with cyclic factors. |

## Latest revision as of 23:27, 5 March 2018

A group having a normal series

$$G=A_1\supseteq A_2\supseteq\ldots\supseteq A_{k+1}=\{1\}$$

such that every quotient $A_i/A_{i+1}$ lies in the centre of $G/A_{i+1}$ (a so-called *central series*). The length of a shortest central series of a nilpotent group is called its class (or degree of nilpotency). In any nilpotent group the lower (and upper) central series (see Subgroup series) breaks off at the trivial subgroup (the group itself), and their lengths are equal to the nilpotency class of the group.

The finite nilpotent groups are exhausted by direct products of $p$-groups, that is, groups of orders $p^k$, where $p$ is a prime number. In any nilpotent group the elements of finite order form a subgroup, the quotient group by which is torsion free. The finitely-generated torsion-free nilpotent groups are exhausted by the groups of integral triangular matrices with 1's along the main diagonal, and their subgroups. Every finitely-generated torsion-free nilpotent group can be approximated by a finite $p$-group for every prime $p$. Finitely-generated nilpotent groups are polycyclic groups (cf. Polycyclic group) and, moreover, have a central series with cyclic factors.

All nilpotent groups of class at most $c$ form a variety (see Variety of groups), defined by the identity

$$[[\ldots[[x_1,x_2]x_3],\ldots]x_{c+1}]=1.$$

The free groups of this variety are called free nilpotent groups. Concerning completions of torsion-free nilpotent groups see Locally nilpotent group.

#### References

[1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |

[2] | M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) |

#### Comments

Let $G$ be a group and $R$ some relation (or, more generally, a predicate) that can hold between elements of groups and/or between sets of elements of groups. For instance, $R$ could be the relation of equality, the relation "the element g belongs to the subgroup H" , or the relation of conjugacy between subsets of a group. Let $\mathcal C$ be a class of groups. Then one says that the group $G$ is approximable by the groups from $\mathcal C$ with respect to the relation $R$ if whenever the relation $R$ does not hold in $G$ (between elements, subsets, or between an element and a subset), then there is a homomorphism from $G$ into a group from $\mathcal C$ such that the relation $R$ also does not hold between the images under this homomorphism. I.e. the homomorphisms $G\to C$, $C\in\mathcal C$, suffice to detect whether elements (subsets) are $R$-different.

If the relation $R$ is equality, then one simply says that the group $G$ is approximable by the groups from $\mathcal C$.

Concerning approximability cf. also Residually-finite group.

**How to Cite This Entry:**

Nilpotent group.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Nilpotent_group&oldid=31728